-fiD-fli38 544 RAVLEIGH-TAVLOR INSTflBILITV IN THE PRESENCE OF fi 1/i
STRATIFIED SHEAR LAVEROJ) NAVAL RESEARCH LAB WASHINGTON
DC P SATVANARAVANA ET AL. 16 FEB 84 NRL-MR-5262
F/G 4/1
UNCLASSIFIED
NL
AD A 1 3 8544
NRL Memorandum Report S263
Rayleigh-Taylor Instability in the Presence of
a Stratified Shear Layer
P. Satyanarayana and P N. Gi zdar
Science Applications. Inc.
McLean. VA 22102
J. D. Hi;ba and S. L. Ossakow
Plasma Physics Division
February 16. 1^84
This research was sponsored by the Defense Nuclear Agency under Subtask S99QMXBC,
work unit 00067, work unit title “Plasma Structure Evolution," and by the Office of
Naval Research.
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P. Satyanarayana,* P.N. Guzdar,* J.D. Huba and S.L. Ossakow
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IS. SUBJECT TERMS ^Continue on mveree If necemary end identify by block number >
Rayleigh-Taylor instability
Ionosphere
Velocity shear
Nonlocal theory
IS. ABSTRACT (Ceelmite on mwerae if neeeeeery and te stify by block number 1
>A nonlocal theory of the Rayleigh-Taylor instability which includes the effect of a transverse velocity shear is presented.
A two fluid model is used to describe an inhomogeneous plasma under the influence of gravity and sheared equilibrium flow
velocity, and to derive a differential equation describing the generalized Rayleigh-Taylor instability. An extensive parametric
study is made in the coilisionless and collisional regime, and the corresponding dispersion curves are presented. The results
axe applied to the equatorial F region and to barium releases in the ionosphere.
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Ha NAM* OR AtSRONSlSL* INDIVIDUAL
P. Satyanarayana
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11. TITLE
RAYLEIGH-TAYLOR INSTABILITY IN THE PRESENCE OF A STRATIFIED SHEAR LAYER
773T
5T
RAYLEIGH-TAYLOR INSTABILITY IN THE PRESENCE OF
A STRATIFIED SHEAR LAYER
I. Introduction
The Rayleigh-Taylor instability (Rayleigh, 1894; Taylor, 1950) arises in
a wide range of physical phenomena. This instability is primarily driven by a
gravitational force acting on an inverted density gradient (e.g., a heavy
fluid supported by a light fluid). In a magnetized plasma these modes can
exist in both the collisionless and colllsional domains. For example,
theoretical models (Hudson and Kennel, 1975; Ossakow, 1979) and nui leal
simulations (Scannapieco and Ossakow, 1976; Ossakow et al. , 1979; Zalesat. and
Ossakow, 1980; Zalesak et al., 1982) of the colllsional Rayleigh-Taylor
instability in the earth's ionosphere show that the mode evolves into plasma
bubbles that extend upward from the bottomside to the topside of the F-
layer. The collisionless interchange type instability (ballooning mode) can
exist in the earth's plasmasphere (Vines, 1980) as well as in laboratory
plasmas (Coppl and Rosenbluth, 1966; Coppi et al., 1979). These collisionless
• I
modes arise due to an unfavorable curvature in the magnetic field (simulating
an effective gravity) in the presence of a pressure gradient. This
instability may also arise in the acceleration of a heavy fluid by one of
lower density as in targets accelerated by laser ablation (Emery et al. , 1982
and references therein) or the deceleration of barium clouds injected in the
ionosphere (Pillip, 1971; Rosenberg, 1971; Davis et al., 1974; Fedder, 1980).
In some of the above situations, the equilibrium flow velocity is
observed to be inhomogeneous. For example, in the ionosphere, the horizontal
plasma velocity during equatorial spread F (ESF) reverses its direction as a
function of altitude (Kudeki et al. , 1981; Tsunoda, 1981a; Tsunoda and White,
1981; Kelley et al., 1981). In the plasmasphere, steep shear in the flow
Manuscript approved November 1, 1983.
velocity can exist due to the dominating corotating electric field inside the
plasmasphere and a convective magnetospheric electric field penetrating across
the plasmapause. This can lead to a Kelvin-Helmholtz type erosion of the
outer edge of the plasmasphere (Vinas, 1980; Vinas and Madden, 1983). In
targets accelerated by laser ablation, the Rayleigh-Taylor instability
(Bodner, 1974) nonlinearly evolves into a bubble and spike structure (as
during ESF) and developes a strong shear in the flow velocity (Harlow and
Welsh, 1966; Daly, 1967; Emery et al., 1982). In the absence of any other
driving mechanism, the velocity shear gives rise to a transverse Kelvin-
Helmholtz instability in fluids (Kelvin, 1910; Chandrasekhar, 1961) and in
magnetized plasmas (Mikhailovskii, 1974). All of the above examples suggest a
need for a detailed study of configurations in which the two driving
mechanisms co-exist, namely, inhomogeneous velocity flows and gravity (or
similar forces).
The influence of velocity shear on interchange instabilities has been
studied by Drazin (1958) and Chandrasekhar (1961) in the context of fluid
models; by Hamieri (1979) in the context of laboratory plasmas; by Vinas
(1980) in the context of the plasmasphere; and by Guzdar et al. (1982; 1983)
in the context of equatorial spread F. In this paper we report on our
detailed study of the Influence of velocity shear on the collisionless and
collislonal Rayleigh-Taylor instability and apply the results to a variety of
geophysical phenomena. We find that the velocity shear can have a dramatic
effect on the Rayleigh-Taylor instability (Guzdar et al., 1982; 1983). A
sufficiently strong velocity shear can stabilize the most unstable modes
(l.e., those for which kL > 1 where L is the scale length of the Inhomogeneity
and k is the perpendicular wave number), which leads to maximum growth in the
long wavelength regime (for which kL < 1). Thus velocity shear, in some
domains, preferentially excites a long wavelength mode, in sharp contrast to
the behavior of the mode in the absence of velocity shear.
This paper is divided into five sections. In the next section, we derive
the general mode structure equation describing an inhomogeneous collisional
plasma which contains a sheared velocity flow and which is under the action of
gravity. In the third section we discuss the stability of this plasma in two
limits, namely, the Rayleigh-Taylor limit (no velocity shear), and the Kelvin-
Helmholtz limit (no gravity). In the fourth section we present the results of
the analysis of the generalized Rayleigh-Taylor instability, i.e., when both
velocity shear and gravity are present. In the final section, we discuss the
results and apply them to geophysical phenomena.
II. Theory
We consider an infinite slab of magnetized plasma. The coordinate system
is shown in Fig. 1. The magnetic field is uniform and is in
A A
the z_ direction (B * B z), the acceleration due to gravity is in the x
A
direction (g * -g x), the ambient electric field is inhomogeneous and is in
A
the x direction (Eq - Eq(x) x), and the density is inhomogeneous in the x
direction (n^ ■ n^Cx)). The inhomogeneity in the background electric field
A
leads to a sheared equilibrium flow, Vq(x) ■ -c Eq(x)/B £.
The basic assumptions used in the analysis are as follows: (1) the
perturbed quantities vary as 5p ~ 5p(x) exp [i(k^y - tot)], where ky is the
wave number along y direction and to ■ tor + iy , implying growth for y > 0;
(2) the ordering in the frequencies is such that to, << where v is
the ion-neutral collision frequency and is the ion-gyro-frequency; (3) we
ignore finite-gyroradius effects by limiting the wavelength domain to
kpj<< 1, where is the mean ion-Larmot radius; (4) we neglect perturbations
along the magnetic field (kfl * 0) so that only two dimensional mode structure
in the x-y plane is obtained; (5) we retain ion inertia effects, thereby
including the ion polarization drift, but Ignore electron inertia; and (6) we
neglect ion and electron pressure.
A key feature of our analysis is that a nonlocal theory is developed.
That is, the mode structure of the potential in the x direction, the direction
in which density and the flow velocity are assumed to vary, is determined by a
differential equation rather than an algebraic equation obtained by Fourier
analysis. This technique allows one to study modes which have wavelengths
comparable to the scale size of the inhomogeneities (i.e., kyL < 1, where L
represents scale lengths of the boundary layer). In fact, nonlocal theory is
<X
5
crucial to describe the Kelvin-Helmholtz instability driven by transverse
velocity shear (Milchailovskii, 1974).
Based upon the assumptions discussed above, the fundamental fluid
equations used in the analysis are continuity and momentum transfer in the
neutfal frame of reference:
3n
FT + 7 * (na V " °»
0 - £ + I (2)
mi(!t + li* 7) li " ei + | (li x - Vin -t + mi 4* (3)
where a denotes species (e for electrons; i for ions).
The equilibrium velocities are, for electrons
IeO(x) 5 Vx) " (c/8> Vx) * 2 » (*>
and for ions (to order v. /fl )
in 1
• (c/B) i [e„ +(B1/e)gJ » £ + (»ln/ Slj)
+ (!«,) (ViQ- 7) Ej
where * eB/m^c . The electrons simply _E x ]J drift, while the ions drift
with which incorporates the x B_ drift, and the effects of gravity and
the polarization electric field.
We substitue E - - 7$ » - 7 ($o<x) + 5$) , - V^Cx) + fiV^ ,
n ■ n^(x) + fin into Eqs.- (1) - (3). To obtain the perturbed velocities we do
not Fourier analyze the perturbed quantities in the x direction since the
equilibrium quantities vary in that direction; we Fourier analyze only in the
y direction. Linearizing Eqs. (2) and (3) and making use of Eqs. (4) and (5),
we find the perturbed velocities to be
where
and
$ « - (c/B) file 6* x + ^ 6* Z],
s'i
TT l' V* +
iil) - v
Q,
+ (c/B) [fj 5* -
to + iv
in
a> « w - kyVQ(x) and VQ - - -(c/B) E^.
We substitute Eqs. (6) and (7) into Eq. (1), to obtain
It «n + i K Vn «» - It k «♦ nn - 0
3t e
y 0 e B y
(6)
(7)
(8)
It 5ni * r??.nof(" ia> + vin + Vo5 (r? " ky ) 6*
i d X
+ 1 ky V
-f iky - Vai> “8 5*
- ra^* 1 “ + vin + 1 kyV no h 5+ (9>
+ (8 /at - v0) iy.j + <vin/n±) (g/nr v0> fj
6n.
+ -si[(g/n, - v„) v;_ + v, v: ] -
o
where we have ignored terms of order 0(oj/0^) but have retained terms
proportional to and the derivatives of the equilibrium quantities with
respect to x (indicated by the primes). We now subtract the electron
continuity equation, Eq. (8), from the ion continuity equation, Eq. (9), and
impose quasineutrality (5ng ■ Sn^). We use Eq. (8) to eliminate 5n^ and
obtain the mode structure equation
a2 a
— j <S$ + p <5$ + q 5$ ■ 0,
3x
(10)
where
n' iv.
0 , in
p . - — + — - -
n0 ( u + iv. ) vin
In
Xln (n0/n0)kyV0 ,
',\)J ^ '
0)
(11)
.2 Vro<W + V> .N * |«
q ■ - k + — ■ — . ' ... .... + i./ .
y “ “ “
(M +lvin)
<*>(<*> + iv±n)
iv. k Vn
in y 0
u> (a» + iv. ) u)
in
t-!- <W<VV + ‘"'o'V + <^n/',l„)(n'o/nO)! <12>
Note that the solution of Eq. (10) allows for wavepacket formation
instead of plane waves. The properties of the wavepacket are governed by the
coefficients p and q. From the expression for q we see that the second term
involves the free energy associated with the inhomogeneous plasma flow giving
rise to Kelvin-Helmholtz instability. The third term has the gravity and the
inverted density gradient leading to Rayleigh-Taylor instability. The last
term becomes Important in the moderately collisional domain.
Now we define the following dimensionless parameters in order to cast Eq.
(10) into a dimensionless form
,1/2
<*> " a>/(g/L)A/i, v - vin/(g/L)1/2, VQ - V0/(gL)i/Z, and k - k L (13)
1/2
where L is the characteristic inhomogeneity scale length. We also normalize x
with L and define a new independent variable x = x/L. With these definitions,
Eqs. (10)-(12) become
~2 + P lx" ^ + 3 m °>
(14)
* % (nl/n.) k V_
P - (p0/p'J ♦ 1 -S- [ 4 - °-°I - — ],
ui2 v (o^
(15)
u>2
. -2 + i { jjo <VV * Vi , }
(Oj u>2
- k ^ + (n0Vn0) + (no/no)(v'/v)]
(Oj o>2
^ a a a
(16)
A A A A A A A
where toj* <o - k V^(x) and u>2 ■ + i v.
We use the transformation
5<Kx)
’Kx) e
r
-J p(q) dn
(17)
9
to write the Eq. (14) as
r' + q(x) * - o,
(18)
where
Q(x) ■ q(x) “ P'(x)/2 - P(x)2/4. (19)
In solving the eigenvalue problem, we use WKB boundary conditions on i|> :
1/4
ij» ♦ (1/Q ' ) exp(-/ Q(n) dn ) as k + ± • . (20)
0
We refer to Q(x) as the potential function and we give the plots of Q(x)
and t|; when we solve Eq. (18) numerically in the next section.
Equations similar to Eq. (18) have been obtained by several authors
studying the stability of stratified shear layers in neutral fluids (Drazin,
1958). However, usually a Rayleigh-Taylor stable density profile was chosen,
mainly to examine the influence of inertia on the velocity shear induced
mixing phenomenon. This situation differs slightly from the one considered in
this paper, since we study a situation where inverted density gradients and
velocity shear both are sources of free energy. The collisionless case can be
A
compared with the neutral fluid case, and when we set v ■ 0, in Eqs. (14)-(16)
we regain the mode structure equation obtained by Drazin (1958). Drazin
(1958) considers a weakly inhomogeneous plasma, i.e., by setting n^/n^ to zero
everywhere except in the driving term containing the gravity, and obtains a
simple equation
This implies that for R < 1/4 the system is stable. Hamieri (1979) considered
a more general case applicable to a Tormac machine and arrived at a less
stringent condition. A theoretical analysis in the collisionless and
collisional cases will be presented in a future paper.
III. Analysis and Results
The generalized mode structure equation [Eq. (14)] can be better
understood by first considering two limiting cases: (A) the collisionless and
collisional Rayleigh-Taylor instability without velocity shear, and (B) the
Kelvin-Helmholtz instability with no collisions or gravity.
A. Rayleigh-Taylor Instability
Setting Vq ■ 0 in Eq. (14) we obtain
r' + r - k2 [i - - 0 . ] * - o (24)
a) (ui + iv )
Eq. (24) can be solved in the local approximation,
>|>(x) * exp(ik^x) ; k2L2 » k2L2 » 1 (25)
and we obtain the well known dispersion relation (Haerendel, 1974; Hudson and
Kennel, 1975)
iv. (I)
in
+ g/L
(26)
which has the solution
CD
[1 + ( 1 - 48/Vin)]1/2’
(27)
-1 1 0
where L ■ — ^ — . Instability can occur when g/L„ < 0. The collisionless
n ng a x n
and collisional solutions are, respectively,
•> - - (8/^)
“ ■ - 1 (*/ln>/vl„
v,„ « 12(8/1..)*' 2I,
uin » I2(8/V1/2'
We now solve Eq. (24) numerically for a density profile
im
nQ - nQ exp(- -^f-) + An.
The results are shown in Fig. 2, which is a plot of normalized growth
rate, y ■ y/(g/L) , vs normalized wave numbers, k - k^L. Curve A is for
the collisionless Rayleigh-Taylor instability and curve B is for the
*
collislonal Rayleigh-Taylor instability with v ■ 0.5. We use An/n^ » 0.01 for
a
both cases. As expected, the growth rate maximizes in the regime k » 1 and
the maximum growth rate agrees well with the growth rate predicted by local
theory [Eq. (27)] with the growth rate evaluated at the maximum density
gradient. The potential function (Q(x) given by Eq. (19)) and the wave
A
function (iji(x)) » corresponding to k ■ 1.0 are shown in Figs. 3 and 4,
respectively. We note that for An/n^ - 0.01, n'/n^ has a maximum at
X ■ -2.4and a potential well [-Q(x) is a minimum] is formed around this point
as can be seen from Fig. 3. We see from Fig. 4 that the wave function also
localizes at x “ -2.4. The negative sign implies that the Rayleigh-Taylor
instability is active where the density gradient opposes the gravity. We note
that the wave function spreads out into the positive region of the x-axls,
where the Rayleigh-Taylor instability is locally stable (gravity acts in the
same direction as the density gradient).
% • v* •_# * * *
Normalized growth rate Y = Y//g /t vs k = kyL for the Rayleigh-
Taylor instability. Curve A represents the collisionless
A
mode (v • 0); curve B represents the colllsional mode
A A ^
(v - 0.5), where v - v//g/L . The density profile used is a
Gaussian-like profile nne-*2^^ + An, with An/nQ ■ 0.01.
B. Kelvin-Helmholtz Instability
We retain the flow velocity Vq ■ Vq(x) but consider a collisionless,
uniform fluid with no gravity. Eq. (14) becomes
k V"
y o
Y' + [- k + 7 — T „
Y 1 y (oi - k^V
-I - 0,
y 0
(31)
which is well known (Mikhailoviskii , 1974). Rayleigh's theorem
(Mikhailovskii, 1974) predicts an instability if the velocity profile has a
vanishing second derivative between the boundaries, i.e., [92V /3X2] * 0,
u x0
where x^ < Xq < %2 an<^ and x2 are c^e boundaries.
Equation (31) is solved for an equilibrium velocity profile
Vq • VQ tanh(x/L) y
(32)
and the results are shown in Fig. 5 (curve A) in which we plot
y/(Vq/L ) versus kyL. The instability is purely growing and is bounded
between kyL * 0 and 1 with a maximum growth rate of y * 0.18 (Vq/L) at
k^L « 0.45 (Michalke, 1964).
When a density gradient is included, we arrive at the following equation
*" + <no/no) + t “ K
IV + ro w
< “o- W
] \)i * 0.
(33)
Using the same pocedure as outlined following Eq. (31), we can show that
for instabilty the density and velocity profiles should be such that
1 3
[— (n0^o ^x * 0 where xq is any point within the boundaries. It is
interesting to note from Eq. (33) that no instability exists if the density
and velocity profiles are such that 9 "^o^O
Equation (33) is
17
IV. Generalized Xayleigh-Taylor Instability
In the previous section we considered the limiting cases where an
inverted density gradient in the presence of gravity and a velocity shear |
individually give rise to different instabilities. We now consider the
general problem where both free energy sources jointly give rise to a
generalized Rayleigh-Taylor instability (Hamieri, 1979; Vinas, 1980). i
We consider two different cases: (A) a self-consistent equilibrium, and
(3) a general equilibrium based on the experimental observations.
A. Self-consistent Equilibrium j
We choose the following density profile
\
nQ(x) - nQ (1 +e tanh(x/L))/(l-e), (34) j
and the following velocity profile
I
V0(x) - V0(n0/n0(x)) (35)
such that
‘ V0 "0 - n0
0 " (gL)1/2 V*’ ' * n0(x) ’
where we have defined a dimensionless parameter s = /(gL) . Note that
the zeroth order continuity equation is satisfied by these profiles for
v. constant, i.e.,
in
[(nn(x) ] = 0.
(36)
which implies
Hq(x) Eq(x) * constant
or using the definition of Vg from Eq. (4),
ng(x) V^(x) = constant.
We solve Eq. (14) numerically using these profiles and present the results
below.
In Fig. 6 we plot the normalized growth rate v versus k for the
collisionless and collisional cases. The solid lines represent the
A
collisionless case (v * 0) and the dashed lines represent the collisional case
A
( v ■ 0.5). We set e * 0.8 in the density profile. Several points are to be
noted in this figure. First, we note that in a shear-free, collisionless
Rayleigh-Taylor plasma the growth rate asymptotes to the local growth rate
evaluated at the peak density gradient, (solid line), i.e.,
y(lc »1) - [g(n- /n0)iW _ . (37)
X = X
For the density profile given in Eq. (36) n'/nn maximizes at y obtained from
Growth rate y vs k for the generalized Rayleigh-Taylor
instability. Solid lines represent the collisionless case
A a
(v ■ 0.0, s =* 0.0, and 1.0); dashed lines represent the
A A
collisional case (v = 0.5, s ■ 0.0, and 1.0). The profiles used
are the self-consistent profiles, Eqs. (34) and (35).
For e = 0.3 Eq. (33) yields ;< = “0.55 and using this in Eq. (37) we find the
local growth rata to be y 3 1.0. this agrees with the growth rate for large
k (see Fig. 6, solid line). Second, we find that ion-neutral collisions have
a stabilizing influence as seen from the dashed line, which represents the
growth rate curve for v = 0.5 and s = 0. Third, in a collisionless Rayleigh-
Taylor
uns table
plasma
, for s = 1.0 (shear
frequency ,
Vq/L , equal
to / g/L)
velocity
shear
stabilizes the short
wavelength
modes (solid
A
line; s
* 1.0); the
cut-off
: mode number, where the
growth rata
becomes zero.
is k - 11.0. As a result the growth rate maximizes at k - 1.5 and has a
c °
maximum value y = 0.675. Fourth, in a collisional plasma also,
with v - 0.5 the short wavelength modes are completely stabilized (dashed
line; s = 1.0). The cut-off mode number in this case is, k ~ 10.0, which is
c
less than that of the collisionless case. The peak growth rate is also
A A
smaller with y = 0. *5 occurring at k = 1.5. We see that the ion-neutral
ul
collisions not only reduce the growth rate but also reduce the cut-off mode
number.
In Fig. 7 we give the plots of the wavefunction for s => 0, k = 0.5,
/x
v = 0.5 , and s = 0.3. We note that the wavefunction localizes at
Xq = -0.55 which is the point where the density gradient (n^/n^) has an
extremum. The wavefunction localizes at this point because 0<x) has a local
minimum. We refer to this point (for s = 0) as the Rayleigh-Tavior
localization point. The negative sign indicates that the density gradient has
to oppose the gravity for instability.
In Fig. 8 we give the wavefunction for the case s * 1 with the
same v, and e as in Fig. 7. The solid and the dotted lines reoresent the
in
real and imaginary parts of the wavefunction, respectively.
22
We note that when
V ■%
the velocity shear is introduced into the problem, the wavefunction picks up
an imaginary part. Furthermore, we find that for k > 1, the wavefunction
localizes at a point closer to the origin in the velocity shear layer.
In comparing our results with Drazin (1958) we note that since we use a
density profile whose density gradient is not a constant, and since we have a
Raleigh-Taylor unstable plasma, our threshold condition on R is quite
different. This aspect will be dealt with in a future paper.
3. General Equilibrium
Recent experimental observations, made during equatorial spread F (ESF)
(Xudeki et al., 1981) and in the high latitude ionospheric F region (Kelley et
al., 1978), indicate that ionospheric plasmas usually support inhomogeneous
equilibrium plasma flows. In the case of ESF it was found that the flow
velocity reverses its direction as a function of altitude (x, the direction of
the density gradient). Furthermore, the velocity reversal point moves up as
the spread F develops. This equilibrium situation, where the flow velocity
profile is not related to the density profile in a simple manner, is generated
by the coupling of the plasma to the neutral atmosphere, for example, by the
neutral winds and the inherent shear in the neutral wind velocity or in the
case of ESF due to an incomplete coupling caused by background ionospheric
Pedersen conductivity away from the equatorial plane’ (Zalesak et al., 1982).
Our earlier numerical results indicate that the inhomogeneity
in a) - kyVg(x), and not necessarily the and V' ' , is primarily responsible
for stabilizing the short wavelength interchanger modes (Huba et al. , 1983).
Therfore, based on the experimental observations (Kudeki et al. , 1981; Tsunoda
et al., 1981) and our numerical results (Huba et al. , 1983), we choose the
following density and velocity profiles for a general equilibrium study:
n0(X) - nQ (1 + t tanh x ) / ( 1 “ e)
(39)
V0(x) * Vo tanh (x - xQ) (*0)
where Xq is the velocity reversal point (in the ionospheric case, Xq is the
point where the westward flow becomes eastward). Using these profiles we
solve Eq. (14) numerically. Also, for simplicity, we choose the density
gradient scale length and velocity shear scale length to be equal. The
numerical results are given below.
First we study the role of Xq on stability and determine the optimum
Xq to be used in later calculations. We set v = 0.5 and e * 0.8. In Fig. 9
we plot the normalized growth rate y versus k. Curve A shows the nonlocal
collisionless Rayleigh-Taylor instability (s * 0). Curves B, C, and D
correspond to s * 1 for different values of Xq* Curve B gives the growth
A
rates for 3*1 and xQ * “2.0 which shows a significant reduction in the
A A
growth rate. The growth rate maximizes with y * 0.52 at 1: - 3.0. However,
when Xq seC co "O’ 35 (the Rayleigh-Taylor localization point) the growth
rate is sharply reduced, maximizing at k ~ 0.5 with y » 0.256 (curve C). For
Xq * 0 (curve D) there is a significant reduction in the growth rate and a
severe reduction of the k domain for instability. The instability is bounded
a a
between k * 0. 5 and 3.6. Here, the growth rate peaks around k ~ 1.7.
For Xq * 2.0 (not shown) the growth rate curve is similar to that of
Xq * -2.0 (curve C). From this we conclude that the effects of velocity
shear are strongest when the velocity reversal point falls in the Rayleigh-
Taylor localization region.
26
'.--•■-'.y-.y VC-. y..--- -V -w --v-. a -v »j».- ^
-v.
Vv
V *■ Vgtanh(x * Xg) for Xg* “0.5, “2.0, and 0.0. Parameters used
A A
are s * 1, v ■ 0.5, e ■ 0.8, and for the profiles given in Eqs.
In order to throw some light on the variation of the dispersion curves as
a function of the velocity shear we plot y versus k for various values of
s keeping v, z , and fixed at 0.5, 0.3, and -0.55, respectively, in Fig.
10. This figure shows that the general (non self-consistent) profiles yield
results similar to those of the self-consistent profiles (sec. IV. A; see fig.
6). As s is increased, k^ the node number at which the growth rate maximizes,
moves towards smaller k and the growth rate is substantially reduced. For
a
very large shear, s >> 1, the mode becomes purely Kelvin-Heimholt? like,
preferentially exciting a long wavelength mode (k ~ 0.45 ) with the cut-off
A
k less than 1.0 (to be compared with Fig. 5). This aspect is further
A A
illustrated in Fig. 11 where we plot Y versus s for several values of
k. The figure shows that for v * 0.5 , modes with k < 0.3 are always
unstable. Mo amount of shear (measured in units of s ) stabilizes these
modes due to the onset of the Kelvin-Helmholtz instability for these large s
and small k. Furthermore, for the parameters used in the figure, moderate to
A
strong velocity shear stabilizes modes with k > 0.8. An empirical estimate of
A A
the shear that stabilizes the smallest k mode can be obtained from the k =
0.8 curve in Fig. 11, i.e., s = 2.5 or Vq/L ~ 2.5 / g/L; the figure also shows
that the critical shear depends on the wavenumber.
From Fig. 10 we see that for v ■ 0.5, the mode with k ■ 0.45 is the
fastest growing mode for large s. It is interesting to study the behavior of
A A A ^
this mode as a function of s. In Fig. 12 we plot y (for k * 0.45) versus s.
Curves A and B represent the collisionless (v =■ 0) and collisional
a A A
(v * 0.5) cases respectively. Note that for s * 0, y is purely Rayleigh-
A A
Taylor-like; but as s increases, y initially decreases, which shows that
velocity shear is reducing the growth rate of the Rayleigh-Taylor
A
instability. 3eyond s > 1 the velocity shear dominates and the Kelvin-
.n.V.'.'V.^'.V.VA -.Y.V.v
• A_WW« •_* 4* •_* h. *■ ■ » . •» * ^
28
M*J
SWSv
Study of y versus s, for k * 0.45 where the Kelvin-Helmholtz
instability has maximum growth rate. Curves A and B refer to
*
y ■ 0.0 and 0.5 respectively.
Helmholtz mode sets in. We note that Y increases linearly with s for large s
since it is normalized to /g/L and not V^/L . For the collisional case we see
that the velocity shear has a stronger influence over a broader domain in
s and the Kelvin-helmoholtz type instability sets in for larger (s > 2.5).
We see from Fig. 10 that the cut-off mode numbers and the mode numbers
where the growth rate maximizes vary significantly as a function of velocity
shear. In order to show the values they asymptote to for large velocity
shear, we plot the cut off mode numbers k (curves A), and the mode numbers of
the fastest growing modes, k (curves B) as a function of s in Fig. 13. We
m
A
use Xq 3 -0.55 , s = 0.8, and v = 0 and 0.5. Solid lines represent the
collisionless case (v * 0) and dashed lines represent the collisional case
(v * 0.5). From the figure, we see that k^ , and k^ fall sharply as s is
A A
increased and asymptote to smaller k values. For v = 0, (solid lines)
A A A
k^ asymptotes to 0.5 and k asymptotes to 1.0. 3ecause for large s the mode
A A
is Kelvin-Helmholtz-like, k and k ,as expected, attain the values shown in
m c
A a A
Fig. 5. For v ■ 0.5 (dashed lines) both k^ and kc are initially smaller than
those for the collisionless case. However, as s is increased these maximum
and cut-off wavenumbers achieve a minimum value, then rise and again asymptote
to similar values as those corresponding to the collisionless case, namely 0.5
and 1.0, respectively.
Finally, in Fig. 14 we show the effects of introducing a spatially
dependent collision frequency. In the ionosphere the ion-neutral collision
frequency decreases exponentially as a function of the altitude. We use the
A
profile v ■ 0. 5exp( -x/L) , choosing the scale length to be the same as the
density gradient scale length for simplicity. Curve A shows the growth rate
A A
curve for constant collision frequency, v M 0.5, and for s « 1.0, and
A A
£ » 0.8. The growth rate maximizes at Y a 0.256 around k = 0.55. Curve B,
with v * v(x), shows a drastic reduction in the growth rate. The maximum
growth rate (y * 0.09) occurs at k ~ 0.7. Interestingly, in this case the
lowerbound of the instability is shifted. The domain of unstable wave numbers
is 0.15 < k < 1.2; whereas, for the constant collision frequency case the
domain was 0 < k < 1.2 (the lower bound for curve A is not shown in the
figure). Curve C for a weaker shear, namely s =* 0.5, shows that the maximum
growth rate is comparable to that of curve A. However, in this case modes
with wave numbers k > 3.0 are completely stabilized.
Plots of k (cut-off mode numbers) and k (mode numbers where
c m
A
the growth rate maximizes) as a function of s . The solid and
dashed lines correspond to the collisionless and collisional
A
(v * 0.5) cases respectively. Curves A and S refer to
A A
k and k , respectively.
Figure 14. Normalized growth rate versus the normalized wavenumbers for the
case of a spatially dependent collision frequency. The density
and velocity profiles are given in Eqs. (39) and (40). The
A
parameters used are s * 0.8, Xq = -0.55, and v = 0.5. Curve A
refers to constant collision frequency, v = 0.5, and s * 1.0.
a A
Curves 3 and C refer to v = 0.5 exp(~x) and for s = 1.0 and 0.5
respectively.
V.
Discussion and Conclusions
We have investigated the influence of velocity shear on the Ravleigh-
Taylor instability. The Ravleign-Taylor instability is driven by gravity and
an inverted density gradient. In general this instability is most unstable in
the short wavelength domain, k L > 1, where L is the density inhomogeneity
scale length and k is perpendicular to the density gradient and the magnetic
field. We obtain the well known results that the maximum growth rate is given
by / g/L (g/v.. L) in the collisionless (collisional) domain. On the other
hand, a sheared transverse velocity drives the Kelvin-Heimholtz instability in
the long wavelength domain, kvL < 1. In the presense of transverse velocity
shear, the short wavelength spectrum (k L > 1) of the Ray leigh-Taylor
instability is strongly suppressed or stabilized and the growth rate maximizes
in the long wavelength domain (kvL < 1). Thus, velocity shear causes a long
wavelength mode to be preferentially excited; whereas in the absense of
velocity shear the dominant wave mode usually has a shorter wavelength
determined by initial conditions or non-linear processes. This prominant
conclusion had been stated in an earlier paper (Guzdar et. al. , 1982, 1983).
We note that the wavepacket generally localizes in the region where the
density gradient opposes the gravity, which in our case also happens to be the
shear layer. The wave function falls off rapidly away from the localization
region of the Raleigh-Taylor instability, but still has some finite amplitude
in the stable region (where _g.7n is positive). This is due to the global
sampling of the entire density profile.
Another interesting feature of the generalized (including velocity shear)
Rayleigh-Taylor instability is its crucial dependence on the velocity reversal
point. In the absense of velocity shear, the wave function localizes at a
point, say x > determined primarily by the background density orofile. In the
case of the hyperbolic tangent density profile (Eq. 34), the wave function
localizes in the region where the density gradient opposes the gravity. If
the velocity reversal point, Xq (where the y-component of the equilibrium
velocity changes sign), is in the region where the density gradient is
parallel to the gravity, the velocity shear has a generally stabilizing
influence without the characteristic peak in the growth rate vs wave number
curve in the long wavelength domain (Fig. 9). However, when xw “ Xq» velocity
shear reduces the growth rate significantly and moves the peak toward longer
wavelengths, preferentially exciting longer wavelength modes.
Two possible applications of this theory to ionospheric phenomena have
been discussed in a previous paper (Guzdar et al., 1982, 1983). 3riefly, the
major feature of this theory, viz., preferential excitation of a long
wavelength mode, may explain (1) the structuring (1-3 km) of barium releases
which are injected across the magnetic field (Linson et al. , 1980; Wescott et
al., 1980), and (2) the long wavelength (few hundred kms) oscillations of the
bottomside F layer during equatorial spread F (Tsunoda and White, 1981; Kelley
et al., 1981).
The shaped barium release experiment (Wescott et al. , 1980) was conducted
at high latitudes in the presence of a pulsating aurora at an altitude of 571
km. Numerical simulations showed that a charge separation induced radial
polarization electric field results in an E. x velocity shear. This shear
layer seems to be located in the region where the density gradient is steepest
(Wescott et al., 1980). Our analysis does not strictly apply in the auroral
environment. However, the basic instability leading to structuring of the
barium cloud is possibly a Rayleigh-Taylor type instability (Pillip, 1971;
Fedder, 1980) due to the deceleration of the cloud (Scholer, 1970). So our
results in the collisionless domain could be applied to this case. For
• k ■
example, gradient scale sizes of 500 a - 1 tan can lead to irregularity scale
sizes of 1.5 - 3 kms with growth rates - 10 " sec .
Kudeki et al. (1981) have shown, from the observations at Jicamarca using
a Radar interferometer technique, that the velocity reversal point moves
upward as the spread-F structures evolve. The position of the F-peak was not
available at the time of these measurements. However, we conjecture that
since the velocity reversal point is at a different location with respect to
the F-peak at different times, the velocity shear induced long wavelength
modulations of the bottom side F-layer may not be apparent at all times, but
may be seen when the velocity reversal point is in the Rayleigh-Taylor
localization region (namely, in the bottom side of the F region).
Tsunoda (1983) recently has shown that the background density gradient
has a scale length of 25 km when long wavelength fluctuations were observed in
the bottom side of the F-layer. No velocity shear measurements, such as the
strength of the shear or velocity reversal point, were available. We point
out that the measured absense of the velocity shear prior to or immediately
after the onset of the wave-like structure is expected because ALTAIR needs
the formation of the bubble and spike structures to measure the plasma
velocities. Despite the lack of shear data in his paper there is resonable
agreement between the data and the theoretical results. However, data on
velocity shear, for example by alternate techniques, are crucial to confirm or
disprove the theory. The question of short circuiting effects by the E-layer,
raised by Tsunoda (1983), needs a closer examination and is not addressed
here.
Similar results were obtained by Vinas (1980) in connection with the
investigations of the erosion of the plasmapause. He conjectured that strong
velocity shear could exist in the plasmapause region and lead to long
38
wavelength irregularities in competition with the ballooning mode type
interchange phenomenon. However, an important difference exists in comparing
Vinas' theory with ours, namely, that g/L is positive in his case, meaning
that the heavy fluid supports the lighter fluid. We also find similar results
in the topside of the equatorial ionosphere, where gravity acts in the same
direction as that of the density gradient (a situation similar to Vinas' case)
and the collision frequency is very small. These results indicate that in a
Rayleigh-Taylor stable plasma the velocity shear could excite Kelvin-Helmholtz
type modes. Thus we can conclude that in the absence of equatorial spread F
if the flow velocity in the topside ionosphere is sufficiently strongly
inhomogeneous, it can induce some large scale irregularities. Figure 5 (curve
B) shows that if sufficient velocity shear exists,
sizes of - 300 km with weak growth rates (10 s )
the weakly collisional topside of the ionosphere.
In conclusion, we have shown that:
(i) Sheared plasma velocity flows can have pronounced effects on the
collisional and collisionless Rayleigh-Taylor instabilities.
Sufficiently strong velocity shear preferentially excites a long
wavelength mode. This result may explain the long wavelength
oscillations of the bottomside F layer during equatorial spread F
and the prompt structuring of injected barium clouds (Guzdar et
al., 1982; 1983).
(ii) Since the wavefunction localizes in the Rayleigh-Taylor unstable
region we expect these long wavelength fluctuations to be seen at
the bottomside of the F-layer
(iii) This phenomenon is most likely to occur when the velocity reversal
point is within the Rayleigh-Taylor localization region (where
gravity opposes the density gradient).
(iv) The generalized Rayleigh-Taylor instability is qualitatively
similar but has quantitatively different properties in the
collisional and collisionless domains (see fig. 12). The cut-off
mode numbers and maximally growing mode numbers are different in
these two cases (see fig. 13).
(v) The properties of the Rayleigh-Taylor instability are similar for
self-consistent as well as for general equilibrium density and
velocity profiles.
(vi) As the velocity shear is increased, the cut-off mode numbers and
the maximally growing mode numbers asymptote to values similar to
those of the collisionless Kelvin-Helmholtz instability (see fig.
12).
(vii) A spatially dependent collision frequency alters the results
drastically by reducing the growth rate, and by restricting the
band of unstable wave numbers to a smaller region (see fig. 14).
ACKNOWLEDGMENTS
This work has been supported by Defence Nuclear Agency and Office of
Naval Research. We thank P. Chaturvedi for helpful discussions.
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i
p
s-s
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»*.■•■.
rrrrrrr r -'.’-V-V-'. ■ a/ 7~
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■■. *
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(
J
j
45
i
>
,• V
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